Study of noise effects in electrical impedance tomography with resistor networks

TL;DR

This paper investigates how noise impacts the accuracy of electrical impedance tomography reconstructions using resistor network models and optimal grid parametrizations, highlighting the efficiency of these methods under various noise levels.

Contribution

It provides a numerical analysis of noise effects on EIT inversion with resistor networks and optimal grids, demonstrating their superior performance over other parametrizations.

Findings

Optimal grid parametrization yields the smallest estimation variance.

Estimates are unbiased and near the Cramer-Rao bound for small noise.

Regularization balances variance reduction and bias introduction for larger noise.

Abstract

We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum a posteriori estimates of the conductivity, on optimal grids. For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the Cramer-Rao bound. For larger noise we use regularization and quantify the trade-off between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered.